Finite Group DatabasebetaFinite Group Databasebeta| order | 72 |
| isoclinism | isoclinic class of D9 |
| automorphism | Aut(G) = C22×AGL(1,ℤ/9) (order 216) · Inn(G) = D9 · Out(G) = C22×C3 |
| holomorph | G ⋊ Aut(G) = #72219992 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| direct product | C4×D9 |
| Fitting | C4×C9 ⋊ C2 |
| derived, Sylow | C9 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | D9 |
| commutator, hypocenter | C9 | C2×C4 |
| Frattini, socle | C2×C3 | C2×D3 |
| Fitting | C4×C9 | C2 |
| prime radical | C2×D9 | C2 |
| cosocle | C2×C9 | C22 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C4 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 13 points:
(2,4)(3,7)(5,8)(6,9);(1,6,4,3,8,5,7,2,9)(10,11,12,13)24 conjugacy classes — view the full character table.
| maximal subgroup of | C8×D9, #3663, C2×C4×D9, … |
| maximal normal subgroup of | C8×D9, #3663, C2×C4×D9, … |
| maximal quotient of | C8×D9, #3663, C4×Dic9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(72, 5);